# Module 3 - Equities & Currencies

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In module three, we will explore the importance of the Black- Scholes theory as a theoretical and practical pricing model which is built on the principles of delta heading and no arbitrage. You will learn about the theory and results in the context of equities and currencies using different kinds of mathematics to make you familiar with techniques in current use.

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Black-Scholes Model

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• The assumptions that go into the Black-Scholes equation
• Foundations of options theory: delta hedging and no arbitrage
• The Black-Scholes partial differential equation
• Modifying the equation for commodity and currency options
• The Black-Scholes formulae for calls, puts and simple digitals
• The meaning and importance of the Greeks, delta, gamma, theta, vega and rho
• American options and early exercise
• Relationship between option values and expectations

Martingale Theory - Applications to Option Pricing

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• The Greeks in detail
• Delta, gamma, theta, vega and rho
• Higher-order Greeks
• How traders use the Greeks

Martingales and PDEs: Which, When and Why

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• Computing the price of a derivative as an expectation
• Girsanov's theorem and change of measures
• The fundamental asset pricing formula
• The Black-Scholes Formula
• The Feynman-K_ac formula
• Extensions to Black-Scholes: dividends and time-dependent parameters
• Black's formula for options on futures

Introduction to Numerical Methods

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• The justification for pricing by Monte Carlo simulation
• Grids and discretization of derivatives
• The explicit finite-difference method

Exotic Options

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• Characterisation of exotic options
• Time dependence (Bermudian options)
• Path dependence and embedded decisions
• Asian options

Understanding Volatility

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• The many types of volatility
• The market prices of options tells us about volatility
• The term structure of volatility
• Volatility skews and smiles
• Volatility arbitrage: Should you hedge using implied or actual volatility?

Further Numerical Methods

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• Implicit finite-difference methods including Crank-Nicolson schemes
• Douglas schemes
• Richardson extrapolation
• American-style exercise
• Explicit finite-difference method for two-factor models

Derivatives Market Practice

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• Option traders now and then
• Put-Call Parity in early 1900
• Options Arbitrage Between London and New York (Nelson 1904)
• Delta Hedging
• Arbitrage in early 1900
• Fat-Tails in Price Data
• Some of the Big Ideas in Finance
• Dynamic Delta Hedging
• Bates Jump-Diffusion

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• The names and contract details for basic types of exotic options
• How to classify exotic options according to important features
• How to compare and contrast different contracts
• Pricing exotics using Monte Carlo simulation
• Pricing exotics via partial differential equations and then finite difference methods

Advanced Volatility Modeling in Complete Markets

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• The relationship between implied volatility and actual volatility in a deterministic world
• The difference between 'random' and 'uncertain'
• How to price contracts when volatility, interest rate and dividend are uncertain
• Non-linear pricing equations
• Optimal static hedging with traded options
• How non-linear equations make a mockery of calibration

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Lecture order and content may occasionally change due to circumstances beyond our control. However, this will never affect the quality of the program.

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Quantitative Risk & Return
Data Science & Machine Learning l